OC-IV

1. OC-IV Orbital Concepts and Their Applications in Organic Chemistry Klaus Müller Script ETH Zürich, Spring Semester 2009 Lecture assistants: Deborah Sophie MathisHCI G214 – tel. 24489mathis@org.chem.ethz.ch Alexey FedorovHCI G204 – tel. 34709 fedorov@org.chem.ethz.ch

2. Chapter 1 some basic elements of orbitals see also document:“Basic Features of MO Theory” on http://www.chen.ethz.ch/course/

3. each orbital can be occupied by an electron witha spin (symbol ) or b spin (symbol ). each orbital can be occupied maximally by twoelectrons with anti-parallel spins (symbol ) according to the Pauli principle. yi = ∑ cikfk k Quantum Mechanics neglect of electron correlation Orbital Models free, resting electron E = 0 discrete stationary states for bound electrons (E<0) in an atom or molecule yiei i-th state described by orbital yi electron energy in yi(relative to E=0) given by orbital energy ei spatial distribution of the electron (electron density ri) is given by yi2. approximation of the MO’s by linear combinations of AO’s → LCAO MO approximation for atomsAO’s fk, ek for moleculesMO’s yi, ei

4. rtot = ∑ biri i Typical result of an LCAO MO calculation for the electronic ground state of a molecule with an even number of n electrons using m AO’s for the LCAO MO approximation. … … alternative set of empty localized MO’s (LMO’s) fi* to represent the same virtual electronic space. (m - n/2) antibonding or partially antibonding unoccupied MO’s yi* (ei* > 0). transformation n/2 bonding or partially bonding, doubly occupiedMO’s yi (ei < 0). alternative set of doubly occupied localized MO’s (LMO’s) fi to represent the same electronic structure. … … properties of canonical MO’s properties of LMO’s - delocalized over whole molecule, adapted to the symmetry of the molecule • fully localized, corresponding to • elements of the molecular structure - correct representation of the total electron density distribution rtot total electronic density distribution etot =∑ biei sum of orbital energies as an estimate of relative energies - correct representation of the total orbital electron energy etot i taking into account the interactions between LMO’s construction of approximate LMO’s → orbital ‘conjugative effects’ approximate LMO representation of the molecular electronic structure → qualitative estimates of energy differences between isomeric molecular systems Hybrid-AO method

5. 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 12 12 12 13 13 13 14 14 14 15 15 15 16 16 16 17 17 17 18 18 18 19 19 19 20 20 20 eV eV eV UV-Photoelectron Spectroscopy gives qualitative insight into the molecular orbital structure of closed-shell molecules E Ekin = hn - IPn photoelectron count as a function of Ekin HeI resonance line hn = 21.22 eV e- e = 0(free electron at rest) e1 IP1 ≈ -e1 photo- emission Koopmans’ theorem e2 IP2 ≈ -e2 Frozen Orbital Approximation e3 IP3 ≈ -e3 … gas phase (ultra-high vacuum) sCH/sCC-dom sCC-dom schematic representations of a few canonical MO’s of cyclohexane (symmetry D3d) el.counts/sec sCH/sCC-combined ‘ribbon’ orbitals (eg) sCH-dom sCC-dom p-type pCH2-dominant orbital (a1g) el.counts/sec sCC-dom sCH/sCC-combined orbitals (eu) sCH-dom p-type el.counts/sec sCC-dominant orbital (a1u) UV-Photoelectron Spectra of cyclohexane, cyclohexene, and 1,4-cyclohexadiene P Bischof, J A Hashmall, E Heilbronner, V Hornung, Helv Chim Acta 52, 1745 (1969)

6. canonical LCAO MO’s obtained as solutions from the eigenvalue problem canonical LCAO molecular orbitals - extend over the whole molecule • are symmetric or antisymmetric with respect to all molecular symmetry elements • (note: this is strictly true only for non-degenerate orbitals; however, every set of • degenerate orbitals represents the true molecular symmetry) - are the starting points for the discussion of spectroscopic properties - can be used for the approximate calculation of any other molecular property - are the starting points for calculations of electron correlation effects (going beyond the Hartree-Fock limit) - however, canonical MO’s are quite inconvenient for transparent rationalization of ‘effects’ in organic chemistry LCAO LMO’s by rigorous localization schemes followed by truncation of orbital tails eg a1g 6 equivalent sCC-LMOs eu 6 equivalent axial sCH-LMOs a1u D3d 6 equivalent equatorial sCH-LMOs

7. the LMO representation of the molecular electronic structure and its relations to the elements of the classical structure formula in organic chemistry (a) From the individual sets of occupied and unoccupied canonical MO’s, obtained by rigorous quantum chemical calculations for the electronic ground state of a closed-shell molecule and transformed by unbiased localization schemes, followed by truncation of orbital tails and renormalization, one obtains for eachsingle bond in a classical molecular structure a (s,s*)-LMO pair Eel • unoccupied, • energetically high-lying orbital s* - LMO AB • essentially localized in A-B bond domain, • axially symmetric w.r.t. A-B bond axis • antibonding between atoms A and B: the s*-LMO has a nodal plane • orthogonal to the bond A-B axis A B • doubly occupied • energetically low-lying orbital s - LMO • essentially localized in A-B bond domain, • axially symmetric w.r.t. A-B bond axis AB • bonding between atoms A and B

8. the LMO representation of the molecular electronic structure and its relations to the elements of the classical structure formula in organic chemistry (b) From the individual sets of occupied and unoccupied canonical MO’s, obtained by rigorous quantum chemical calculations for the electronic ground state of a closed-shell molecule and transformed by unbiased localization schemes, followed by truncation of orbital tails and renormalization, one obtains for eachdouble bond in a classical molecular structure a (s,s*)-LMO pair and a (p,p*)-LMO pair Eel • characteristic features of th s*-LMO as in (a) s* - LMO AB p* - LMO • unoccupied, energetically high-lying orbital, • but lying below s* AB • essentially localized in A-B bond domain, • antisymmetric w.r.t. A=B double bond plane • antibonding between A and B, • but less antibonding than s*-LMO; the p*-LMO has two orthogonal nodal planes, • one orthogonal to the bond A-B axis, and • one in the A=B double bond plane A B A=B plane • doubly occupied, energetically low-lying • orbital, but higher than s-LMO p - LMO AB • essentially localized in A=B bond domain, • antisymmetric w.r.t. A=B bond plain • bonding between A and B, • but less bonding than s-LMO • characteristic features of the s-LMO as in (a) s - LMO AB

9. the LMO representation of the molecular electronic structure and its relations to the elements of the classical structure formula in organic chemistry (c) From the individual sets of occupied and unoccupied canonical MO’s, obtained by rigorous quantum chemical calculations for the electronic ground state of a closed-shell molecule and transformed by unbiased localization schemes, followed by truncation of orbital tails and renormalization, one obtains for eachtriple bond in a classical molecular structure a (s,s*)-LMO pair and two mutually orthogonal (p,p*)-LMO pairs Eel • characteristic features of th s*-LMO as in (a) s* - LMO AB • two energetically degenerate • mutually orthogonal p*-LMO’s • characteristic features as in (b) p* - LMO’s AB A B • two energetically degenerate • mutually orthogonal p-LMO’s • characteristic features as in (b) p– LMO’s AB s- LMO • characteristic features of th s-LMO as in (a) AB

10. A the LMO representation of the molecular electronic structure and its relations to the elements of the classical structure formula in organic chemistry (d) From the individual sets of occupied and unoccupied canonical MO’s, obtained by rigorous quantum chemical calculations for the electronic ground state of a closed-shell molecule and transformed by unbiased localization schemes, followed by truncation of orbital tails and renormalization, one obtains for each lone electron pair in a classical molecular structure a doubly occupied n-LMO pair Eel : n- LMO • doubly occupied non-bonding n-LMO A • essentially localized at atom A • energy depending on nature of A as well as amount of valence s-character; • typically higher in energy than s-LMO’s Eel : B : • two energetically degenerate doubly occupied non-bonding n-LMO’s • essentially localized at atom B • energy depending on nature of B as well as amount of valence s-character; • typically higher in energy than s-LMO’s n– LMO’s B

11. AO basis sets in LCAO MO approaches general aspects – STO’s versus GTO’s Based on the exact quantum-mechanical solutions (eigenfunctions) for the H-atom,similar slightly simplified atomic orbitals 1s, 2s, 3s, …, 2p, 3p, …, 3d, etc. can be defined for the heavier atoms. These are Slater AO’s or Slater-type orbitals (STO’s).These AO’s are characterized by an exponential decay (e-ar) of the radial amplitude, which is an important aspect of proper AO’s. However, this results in considerable computational efforts in solving the various integrals involving differential overlaps between AO’s located at different centers. Gaussian functions (e-ar ) are much more convenient. Thus, Gaussian-type orbitals (GTO’s) are typically used nowadays, which results in a massive computational saving. However, for a proper description of the orbital tails more distant from a nucleus, at least two GTO’s of the same type, but different exponentials, have to be combined. This has led to the notion of “STO-nG” which indicates that a linear combination of n GTO’s of a given type are used to represent one STO of this type. 2 minimal basis and extended basis sets The minimal basis set of AO’s includes the 1s-AO for the H-atom and the 1s, 2s, and 2p-AO’s for the first-row heavy atoms. While minimal-basis set calculations can provide a good, albeit rough orientation about the electronic structure of organic molecules, larger basis sets are typically used nowadays. The advantage of minimal-basis set calculations lies in their more direct correlation with structural concepts of organic chemistry. Extended orbital basis sets typically include several GTO’s, a most prominent approach consisting in using a so-called split-valence basis set, in which each valence orbital (e.g., 2s, 2p for the 1st-row heavy atoms) is represented by 2 (valence double-zeta), 3 (valence triple-zeta), or 4 (valence quadruple-zeta) orbitals of the corresponding type. The notation “6-31g” indicates that the core AO’s is approximated by 6 GTO’s, while for the valence AO’s, a double-zeta split is used with 3 GTO’s in fixed combination for the first component of the valence orbital and a single GTO as a variable second component. An increase in the basis set is often balanced with the addition of polarization functions (e.g., p-AO for H-atom, d-AO’s for 1st-row heavy atoms); such basis sets are denoted with an asterisk, e.g., 6-31G*. With a properly balanced increase of the basis set the computational results ultimately converge to the ‘Hartree-Fock’ limit, which can then be used for calculations of electron correlation effects (post-Hartree-Fock calculations). Note that correlation calculations are of little meaning when medium-size or highly unbalanced extended basis sets are used. “Correlation-consistent” extended basis sets have been derived and are typically denoted as cc-pVDZ, cc-pVTZ, where “cc-p” stands for “correlation-consistent polarized”, and VDZ, VTZ, etc., stand for, respectively, valence-double-zeta” and “valence-triple-zeta”.

12. z z z y y y x x x approximate LMO’s by means of the Hybrid-AO approach The Hybrid-AO approach used in organic chemistry is primarily restricted to a minimal valence-AO basis set, i.e.: number of AO’s symbolic representation atom 1s H 1 1st-row(B,C,N,O,F) 4 2s 2px 2py 2pz Such symbols are used throughout. They denote in gross qualitative terms the contour surfaces of constant absolute amplitude for a given AO, with the shading indicating positive amplitude domains. s-AO’s are spherically symmetric, 1s and 2s differing mainly in their radial extensions; however, for reasons of orthogonality to the core-1s AO, the 2s valence AO has a spherical nodal surface, separating a small inner (negative domain) from the dominant outer domain. The nodal aspect of the 2s-AO is ignored in qualitative hybrid-AO treatments. The p-AO’s are axially symmetric with one nodal plane orthogonal to the orbital axis. A p-AO is antisymmetric with respect to its nodal plane. The three p-AO’s are mutually orthogonal. They form a Cartesian vector system, i.e., a p-AO in any specific spatial orientation can be vectorially decomposed into its px-, py-, pz-components. Also note that the three p-AO’s complement each other to the full spherical symmetry, i.e., the combined density distributions of px, py, pz, constitute a spherical density distribution. 2 2 2

13. He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V H 0 3p 4s 2p 3s 2s 3d -10 1s -20 -30 -40 -50 -60 -70 -80 -90 -100 eV core- and (averaged) valence orbital energies from X-ray PES for the free atoms (using Al Ka1,2 X-rays at 1486.6 eV) by D. A. Shirley et al., Phys. Rev. B 15, 544 (1977) Electrons in 2p-AO’s are more shielded by the core electrons from the positively charged nucleus than electrons in 2s-AO’s. Accordingly, 2s-AO are lower in energy than 2p-AO’s. The energy gap DE2p-2s increases with increasing nuclear charge. Similar trends are seen for the 3p and 3s-AO’s of the 2nd-row heavy atoms.The approximate DE2p-2s for C, N, O atoms are, respectively, 8 eV, 11 eV, and 16 eV.